Real structures of Teichmüller spaces, Dehn twists, and moduli spaces of real curves

Peter Buser and Mika Seppälä

An orientation reversing involution  s  of a topological compact genus g,  g>2, surface  S  induces an antiholomorphic involution
 

s^*: T^g  -> T^g

of the Teichmüller space of of genus g Riemann surfaces. Two such involutions  s^* and t^* are conjugate in the mapping class group if and  only  if  the corresponding orientation reversing involutions    s  and  t  of   S are conjugate in the automorphism group of  S. This is equivalent to saying that the quotient surfaces
 

S/<s>    and   S/<t>
 

This result is a simple fact that follows from  Royden's theorem stating that the the mapping class group is the full group of  holomorphic automorphisms  of the  Teichmüller space (g>2).

Let
 

s^*: T^g  -> T^g   and  t ^*: T^g  -> T^g

be two real structures  that are not conjugate in the mapping class group.  In this paper  we construct   a real analytic diffeomorphism d: T^g  -> T^g such that
 

s ^*  =   d ^-1   o   t^*   o  d.

This mapping   d   is a product of  full and half Dehn-twists around certain simple closed curves on the surface   S.

This  result has applications to the moduli spaces of real algebraic curves.